[DWarrior]

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WTF, I'm rusty on my statistics, but are you going to tell me that a .5% difference in VPIP over 100,000 hands is 3 SD's? [censored], post your math.

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(14%)*(1-14%)/sqrt(100000) = .038% = 1SD

Of course it's higher than that because all 100,000 decisions are not independent identically distributed.

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I think the test statistic formula is .005/sqrt(.14*.86/100000) = 4.6

Assuming independence, and I'm just using a textbook formula for population proportions.

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We just went over this last week at work..
(sigma) = squareroot( (sumation((x-change of x)squared) divided by n-1

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I'm not sure what this refers to.

But can't we also use the Goodness of Fit test for this?

We have 4 suspected bots in the OP's screenshot:

VPIP1: 14376/105366 = 0.136438699
VPIP2: 15840/112514 = 0.14078248
VPIP3: 11683/82577 = 0.141480073
VPIP4: 5721/41414 = 0.138141691

The total VPIP is 47620/341871 = 0.139292306

So if they are indeed botting, the expected hands played of the 4 bots should be:
14676.67
15672.33
11502.34
5768.65

respectively

Throwing it into my calculator for GOF, using 3 degrees of freedom, we get a p-value of .0108, which means there's just over 1% chance that this happened by chance if we assume the 4 players were using a similar strategy.